A379019 Positive integers k such that the simplest cubic field defined by x^3 - k*x^2 - (k+3)*x - 1 is not monogenic.

21, 30, 41, 48, 57, 75, 84, 90, 100, 102, 103, 111, 129, 138, 139, 152, 154, 156, 165, 183, 188, 192, 201, 204, 210, 219, 235, 237, 246, 250, 264, 269, 271, 273, 291, 299, 300, 318, 327, 335, 345, 348, 354, 356, 372, 374, 381, 384, 398, 399, 404, 408, 426, 433, 435, 438, 446, 453, 462, 480

Offset

1,1

Comments

These are the positive integers k such that the ring of integers O_K of the simplest cubic field K = Q[x]/(x^3 - k*x^2 - (k+3)*x - 1) does not have a power integral basis of the form {1, a, a^2} for any element a in O_K.

Links

Table of n, a(n) for n=1..60.

D. Gil-Muñoz and M. Tinková, Additive structure of non-monogenic simplest cubic fields, arXiv:2212.00364 [math.NT], 2022.

T. Kashio and R. Sekigawa, The characterization of cyclic cubic fields with power integral bases, Kodai Math. J. 44 (2021), no. 2, 290-306.

D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.

Prog

(Magma)
is_A379019 := function(k)
    R<x> := PolynomialRing(Integers());
    K<a> := NumberField(x^3 - k*x^2 - (k+3)*x - 1);
    return #IndexFormEquation(MaximalOrder(K), 1) eq 0;
end function;
[k : k in [1..1000] | is_A379019(k)];

Crossrefs

Cf. A005472.

Sequence in context: A215965 A225653 A363609 * A336357 A317772 A308339

Adjacent sequences: A379013 A379016 A379018 * A379021 A379022 A379023

Keyword

nonn,new

Author

Robin Visser, Dec 13 2024

Status

approved